Complex Natural Logarithm/Examples/i

From ProofWiki
Jump to navigation Jump to search

Example of Complex Natural Logarithm

$\ln \paren i = \paren {4 k + 1} \dfrac {\pi i} 2$

for all $k \in \Z$.


Proof

\(\ds i\) \(=\) \(\ds \exp \paren {\dfrac {i \pi} 2}\) Polar Form of $i$
\(\ds \leadsto \ \ \) \(\ds \ln \paren i\) \(=\) \(\ds \ln \paren {\exp \paren {\dfrac {i \pi} 2 + 2 k \pi i} }\)
\(\ds \) \(=\) \(\ds \dfrac {i \pi + 4 k \pi i} 2\) Definition of Complex Natural Logarithm
\(\ds \) \(=\) \(\ds \paren {4 k + 1} \dfrac {\pi i} 2\)

$\blacksquare$


Sources